The whole song is originally from Tom Waits, and it described his noisy neighbours he had. The intro was used during the Origin of Symmetry era as a set opener. It's most notable use is on their Hullabaloo gigs and also at their Reading and Leeds gigs.

What the hell is he building
In there?
He has subscriptions to those
Magazines... He never
Waves when he goes by
He's hiding something from
The rest of us... He's all
To himself... I think I know
Why... He took down the
Tire swing from the Peppertree
He has no children of his
Own you see... He has no dog
And he has no friends and
His lawn is dying... and
What about all those packages
He sends. What's he building in there?
With that hook light
On the stairs. What's he building
In there... I'll tell you one thing
He's not building a playhouse for
The children what's he building
In there?

Now what's that sound from under the door?
He's pounding nails into a
Hardwood floor... and I
Swear to god I heard someone
Moaning low... and I keep
Seeing the blue light of a
T.V. show...
He has a router
And a table saw... and you
Won't believe what Mr. Sticha saw
There's poison underneath the sink
Of course... But there's also
Enough formaldehyde to choke
A horse... What's he building
In there. What the hell is he
Building in there? I heard he
Has an ex-wife in some place
Called Mayors Income, Tennessee
And he used to have a
consulting business in Indonesia...
but what is he building in there?
What the hell is building in there?

He has no friends
But he gets a lot of mail
I'll bet he spent a little
Time in jail...
I heard he was up on the
Roof last night
Signaling with a flashlight
And what's that tune he's
Always whistling...
What's he building in there?
What's he building in there?

Montagues and Capulets - also known as Dance of the Knights - was used as a playback introduction to Knights of Cydonia during the 2007 stadium leg. It is a classical piece with a predominant string section. You may recognise it as the theme tune to the BBC's The Apprentice. It can be seen here, along with Knights of Cydonia.
You can find a live version here. (Flac format, Live at Nimes Arena)

The piece returned as intro music for the European leg of the Drones World Tour (excluding their performances in Paris), where it was used to acompany the arrival of the Drones police right before the show starts (watch)

JFK Speech

John F. Kennedy's speech appeared as an intro at some gigs in 2007 as heard in the clip above.

In 2015, Muse bought the rights to use the song in the album Drones as an intro to Defector, with a string arrangement playing in the background. They titled it JFK. The speech was shortened, as opposed to the 2007 version.

It was later used as an audio filler inbetween the sets on the gigs in 2015 and 2016, being an intro to Uprising, Revolt, Reapers, Stockholm Syndrome and only one performance of Defector.

Like the 2007 version, the visuals display the spoken words.

Lyrics»

The very word "secrecy" is repugnant in a free and open society

And we are as a people inherently and historically
Opposed to secret societies, to secret oaths and to secret proceedings.

For we are opposed around the world

By a monolithic and ruthless conspiracy
That relies primarily on covert means
For expanding its sphere of influence

On infiltration instead of invasion
On subversion instead of elections
On intimidation instead of free choice

It is a system which has conscripted
Vast human and material resources
Into the building of a tightly knit
Highly efficient machine that combines military, diplomatic
Intelligence, economic, scientific and political operations.
Its preparations are concealed, not published.
Its mistakes are buried, not headlined.
Its dissenters are silenced, not praised.

No expendature is questioned
No secret is revealed.
That is why the Athenian lawmaker Solon
Decreded it a crime for any citizen to shrink from controversy.

I am asking your help
In the tremendous task of informing and alerting the American people.
Confident that with your help
Man will be what he was born to be: Free and independent.

We Are the Universe is an intro piece used during The Resistance arena and stadium tours.

Two versions were made. Version one, for indoor arena shows, consists mostly of synth arpeggios and a slowed-down version of the synth in Uprising. Version two, for outdoor shows (stadium and festival gigs), consists of riot noises of protestors, police and pyro/bombs. This second version is also known as "Riot Intro".

Both versions include the following lines (in version one they're spoken by a female robotic voice, while in version two they're spoken by a protestor yelling them in a megaphone):

Lyrics»

We are

the universe
observing itself
observing itself

We are
the universe
observing itself
observing itself

We are
the universe
observing itself
observing itself

We are
the universe
destroying itself
destroying itself
destroying
itself

The 2nd Law of Thermodynamics

Theis intro was spec features speech about the second law of thermodynamics taken from The 2nd Law: Unsustainable. It was specially modified with the speech and the robot voice isolated from the song. There is only the first verse of the speech and the first "Unsustainable" pronounced by the robot, then there is an explosion that leads to Supremacy, which has a longer intro. It was used for the The Unsustainable Tour in 2013, for the Stadium gigs. It is also referred as the "Bomb Intro".

Lyrics»

All natural and technological processes

proceed in such a way that the availability
of the remaining energy decreases.

In all energy exchanges, if no energy
enters or leaves an isolated system,
the entropy of that system increases

This is a very 80s sounding aetheral piece based around an arppegiated synth line.

It is played before Madness. It is used so that the band can take their time getting from the B-stage to the main stage on carts udnerneath the catwalk of the stage. It was also used for shortened setlists at Austin and Singapore before Pressure.

STT Interstitial 3

The third and final of the three Interstitial pieces made specially for the Simulation Theory World Tour in 2019 however it wasn't used up until the third gig.

The synth arpeggio is the same as in STT Interstitial 1, but the song differs in that it features a more prodominent John Carpenter influence with the dark-sounding ambient noise in the background as opposed to a lead synth line playing a melody.

Guitar Noise

A creepy guitar intro where Matt uses effects to make weird sounds with his guitar, before the Metal medley. Before Stockholm Syndrome starts, Matt makes some feedback with his guitar on a D note, that is to say the tuning of the heavy songs that are played next.

It was longer (around 1:40) for the first gigs of the Simulation Theory World Tour, because there was no STT Interstitial 3 yet, but when this interlude started to be used, the Guitar Noise was then shortened (around 20-30 seconds).